Edit: I was not being precise and probably wrong. I thank Denis Nardin for that. I have therefore removed the remarks on $F^\infty$.
Let us consider the invertible bounded linear operators on a (real/complex) Hilbert space $H$. Kuiper's Theorem states that $GL(H)$ is contractible, so all higher homotopy groups vanish. If one considers the subgroup of all invertibles of the form $Id+K$ where $K$ is a compact operator one obtains another group $GL_C(H)$. The homotopy type of this space is $O(\infty)$ or $U(\infty)$ depending if the Hilbert space if real or complex.